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Obtuse Angle: Definition, Degree, Examples

obtuse angle

Geometry deals with shapes and their measurements. When two line segments or rays meet at an inclination, they form a corner called an angle. Angles are classified into various types

  • Acute angle- An angle lesser than 90 degrees is called an acute angle.
  • Right angle- The 90-degree angle is called a right angle. 
  • Obtuse angle- An angle that is greater than 90 degrees but lesser than 180 degrees is called an obtuse angle. 
  • Straight angle- The 180-degree angle is a straight angle.
  • Reflex angle – An angle that is more than 180 degrees but is less than 360 degrees is called a reflex angle.
  • Full angle- The 360 degrees angle is a full angle.

The article below will learn the important concepts related to the obtuse angle. Here is what we’ll cover:

  • What is an obtuse angle?
  • Obtuse Angle Degree
  • Obtuse Angle Triangle
  • Solved Questions on Obtuse Angles

What is an Obtuse Angle?

Obtuse angle definition: In geometry, an angle that is greater than 90 degrees but lesser than 180 degrees is called an obtuse angle. We can easily recognize an obtuse angle because it extends past a right angle.

What does an obtuse angle look like?

A common example to understand what an obtuse angle looks like is to observe the minute and hand hour of a cock. Often during a 24 hours day, a clock frames many obtuse angle degrees between an hour hand and a minute hand. For example, when it is 12:25, the two hands of the clock make an obtuse angle. 

Obtuse Angle Degree

Now we know that an angle that measures less than 180 degrees but more than 90-degrees is an obtuse angle. Some examples of obtuse angle degrees are 110°, 135°, 150°, 179°, 91°, and more. Hence, all angles that lie in the range of 90° to 180° are obtuse angles.

Some real-life obtuse angles that you come across each dayLook around, and you will find a number of objects that depict obtuse angles around you. An open book when you read it, a pair of scissors, a door opened wide are obtuse angles. 

How To Construct An Obtuse Angle?

Two rays meet at a point to form an angle. The two rays form the arms of the angle. Vertex is the point /corner where the arms meet. Here we will learn the steps to draw an obtuse angle using a protractor and compass. 

Constructing an Obtuse Angle with Protractor

The following steps will help you construct an obtuse angle with the help of a protractor.

Step 1: Firstly, draw a straight line with the help of a ruler. 

Step 2: Now, place the protractor’s center at one end of the line.

Step 3: Count from zero and mark a dot at the required angle. The figure shows 129 degrees. 

Step 4: Now, remove the protractor. Join the end of the line to the marked dot with the help of a ruler.

Step 5: Mark the angle and check it again with the protractor to see if you joined the line correctly. 

Constructing an Obtuse Angle with Compass

The following steps will help you construct a 120-degree obtuse angle using a compass and ruler, i.e., without using a protractor.

Steps for Construction of 120-degree angle

Step 1: Draw a ray OA. 

Step 2: Now, take O as the center and draw an arc that cuts through the OA. You can take any radius of your choice and use the same radius throughout the Construction. Mark the point as B where the arc cuts OA.  

Step 3: Next, take B as the center and the same radius as before, draw an arc that cuts the first arc at point C. 

Step 4: Now, take C as the center and the same radius, draw an arc that cuts the initial arc. Mark the point as D.

Step 5: Join OD to get 120 degrees angle AOD.

Steps for Construction of 105-degree angle

The following steps will help you construct a 105-degree angle with a compass and ruler.

Step 1: Draw a ray OA. 

Step 2: Now, take O as the center and draw an arc that cuts through the OA. You can take any radius of your choice and use the same radius throughout the Construction. Mark the point as B where the arc cuts OA.

Step 3: Next, take B as the center and the same radius as before, draw an arc that cuts the first arc at point C. 

Step 4: Now, take C as the center and the same radius, draw an arc that cuts the initial arc. Mark the point as D

Step 5: Now draw rays OE and OF passing through points C and D, respectively. Now, angle AOE is 60 degrees, and angle EOF is 60 degrees.

Step 6: Now, taking C and D as centers, draw an arc between the two rays OE and OF such that the two arcs interest each other. Mark the point of intersection as P. Now, angle AOP will be 90 degrees, and angle POD is 30 degrees.

Step 7: Now bisect angle POD for this mark a point Q where OP intersects the initial arc. 

Step 8: Taking Q and D as centers, draw arcs that intersect. Mark the point of intersection as R.

Step 9: Join OR to get 105 degree angle. Verify the angle using a protractor.

Obtuse Angle Triangle

We know that triangles consist of three angles. When one of the angles is obtuse, the triangle is called an obtuse angle triangle. 

Note: A triangle can have only one obtuse angle because the sum of all the angles of a triangle has to be 180 degrees.

So, when any one of the vertex angles of a triangle is more than 90°, it is an obtuse angle triangle. Furthermore, an obtuse angle triangle can be an isosceles or a scalene triangle. So, an equilateral triangle can never be obtuse because it has angles measuring 60 degrees. Also, an obtuse triangle cannot be a right-angle triangle as per the angle sum property (sum of angles of a triangle = 180 degrees).

Some Properties of An Obtuse Angle Triangle

Here are some important properties of obtuse angle triangles:

  • It is one angle that is greater than 90 degrees.
  • The other two angles are always less than 90 degrees.
  • The side opposite to the obtuse angle is the longest side of that triangle. 
  • When one angle of a triangle is obtuse, the other two angles are always acute. 

How to Determine If a Triangle is Obtuse? 

We can easily determine whether a triangle is obtuse if we know the two angles of a triangle. Using the angle sum property, we will find the third angle and decide whether the triangle is an obtuse triangle or not.

But how to determine if a triangle is obtuse when we know just the sides of the triangle and not the angles. Here we will use inequality in the lines of Pythagorean identity to find an obtuse triangle. 

The triangle is obtuse when the sum of the squares of the smaller sides is less than the largest sides’ square. 

If a, b and c are the three lengths of the sides of a triangle ABC. If c is the longest side, then the triangle will be obtuse only if a2 + b2 < c2

Example Questions on Obtuse Angles

Here are a few examples of obtuse angles with solutions that will further explain the concept of obtuse angles.

Example 1:  Which one of the following angles is 130°.

Answer: The angle B is an obtuse angle because it is more than 90 degrees while it is lesser than 180 degrees.

Example 2: In the following figure, can you name the obtuse angle? 

Answer: The obtuse angles are angle BOD, angle AOC, and angle AOD as they are more than 90 degrees but less than 180 degrees. 

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